Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (3^1/2)(3^3/2)

Verified step by step guidance

1. Identify the expression to simplify: \((3^{1/2})(3^{3/2})\).
2. Recall the property of exponents that states when multiplying powers with the same base, you add the exponents: \(a^m \times a^n = a^{m+n}\).
3. Add the exponents: \(\frac{1}{2} + \frac{3}{2} = \frac{4}{2}\).
4. Simplify the sum of the exponents: \(\frac{4}{2} = 2\).
5. Rewrite the expression using the simplified exponent: \$3^2$ (which has no negative exponents).

Key Concepts

Properties of Exponents

This concept involves rules for manipulating expressions with exponents, such as multiplying powers with the same base by adding their exponents. For example, a^m * a^n = a^(m+n). Understanding these properties is essential for simplifying expressions like (3^(1/2))(3^(3/2)).

Rational Exponents

Rational exponents represent roots and powers simultaneously, where a^(m/n) means the nth root of a raised to the mth power. For instance, 3^(1/2) is the square root of 3. Recognizing and working with rational exponents helps simplify expressions involving fractional powers.

Eliminating Negative Exponents

Negative exponents indicate reciprocals, such as a^(-n) = 1/a^n. The problem requires answers without negative exponents, so understanding how to rewrite expressions to avoid negative powers is crucial for presenting the final simplified form correctly.